A Novel Mathematical Model of Tumor Growth Kinetics with Allee Effect under Fuzzy Environment

: Most of the cancer growth models have described the exponential growth patterns at the very initial stage with low cell population density. Eventually, decreasing the tumor growth rate at higher cell population densities because of deficiency in resources such as space and nutrients. However, recent studies at clinical and preclinical investigations of cancer initiation or reappaearance showed a population dynamics evincing that the growth rate increases as cell number increases. Hence, showing behaviour analogous to cooperative mechanism in the ecosystem and ecological effect called Allee effect. Based on these observations with two arguments i.e. change in initial population and growth rate. In this paper, the novel mathematical model of tumor growth kinetics with Allee effect under fuzzy environment is proposed. In this model the Generalized Hukuhara derivative approach is utilized to solve the fuzzy differential equations. Moreover, it is showen that the change in initial value and growth rate affects the cell density with the Allee effect under the fuzzy environment. Finally the superiority of model has been showen with the help of numerical simulation.

species and therefore, collapse the ecosystem (disturbs body's balance, cause the death of healthy cells and made the body like acidic) called cancer [1,2]. The tumor growth cell mechanism depends upon how potentially cells are capable of availing the facilities and how profitably they construct a mechanism to escape the identification and extinction by the defense system of the body [3]. Kirill et. al. [4] highlighted the four prime characteristic features explaining why the cancer is burdensome to cure as 1) Limited information of cellular and tissue level process. 2) Facing challenge to identify and hit cancer cells as they are clone to normal cells. 3) Due to the rapid evolution of cancer cells, it is very hard to overcome the tumor suppressor mechanism. 4) The clonal diversity in heterogeneous makeup thus, permiting for cancer reappearance after remission frequently.
The application of evolutionary modeling to cancer growth has yielded significant results.
Despite widespread recognition of cancer as an evolutionary mechanism, little research has been done to characterize cancer's ecological dimensions. Our understanding of ecological processes with respect to tumors, in particular, is minimal [4 -24]. The logistic model and Gompertz model have been proved the best fit model in describing the tumor dynamics [25]. An ecological effect called as an Allee effect or inverse density dependence is one of the frequent departure from logistic growth and Gompertz growth models [26,27].

Alley Effect in Tumor Growth
Warder Clyde discovered the Allee effect first in 1930. It states that the allele effect is a biological occurrence to divulge a high correlation between population size or density and the individual mean population fitness. The population with a peak per capita growth rate at low cell population density corresponds to a phenomenon called Allee effect. It happens when the growth rate per capita rises as the density increases and decreases when a certain level is reached, which is often referred to as the threshold. Variations in various parameters, such as social isolation in small-scale, inbreeding depression, finding a spouse, predator avoidance of defence and food exploitation may all contribute to this effect and the problem of searching indo-individual pairs of low population density species is the main cause of this effect [28 -35].
Ecologists and mathematicians have since studied this effect in many different situations in different fields. It is assumed, in most models that the growth of cancers by cell-autonomous proliferation, evinced as an increase in cell number exponentially at initial and finally restricted by the carrying capacity [36]. However, the Allele effect shows active involvement in tumor growth kinetics by describing the cancer growth at low cell population densities deviates from the exponential growth path [37,4,38]. The impact on tumor growth due to the Allee effect through the features of complex models integrating the Allee effect and the implications of the presence of such an effect on the option of the most suitable therapy is described [35].
Allee effect can be induced by several factors mentioned above but among them, potentially relevant to cancer can be a cooperative interaction mechanism [4]. A population that may increase at a medium population density but decreases if the species count is either too small or too large describes strong Allee effect while as a weak Allee Effect represents a population that rises at a low population level with a decreased per capita growth rate, although the growth rate is still positive [39]. A very recent study identified the possible presence of a weak Allee effect in tumor cells. Through a series of in vitro experiments the lower density rate of tumor cell growth was tested with two glioblastoma cell lines, where it was observed that the growth rate increased with population density when the cell density was low while decreasing at higher density. weak Allee effect of cancer cells in cultures probably results from autocrine growth factors, diffuse signaling molecules formed and secreted by cells that promote the growth and proliferation of other cells [40,38]. The mathematical model of a tumor with the week alley effect demonstrates how the intensity of the Allee effect affects the tumor size [41].

Fuzzy Theory and Applications
The fuzzy mathematical modeling can be used to simulate a variety of real-world phenomena where there are certain complexities due to inexactness and vagueness [42 -45].
Firstly, the fuzzy concept in 1956, has introduced by Professor Lotfi Ahmad Zadeh, at the University of Berkeley and the ambiguous and impreciseness are dealt with in fuzzy logic. In other words, rather than the standard true/false or 1/0 like Boolean logic, it provides an understanding of real-life problems dependent on degrees of fact. The concept of fuzzy is logic that is used to designate fuzziness rather than logic that is fuzzy. In oncology, related to tumor growth, fuzzy mathematical models have been used by A.M Nasarbadi in 2009 and 2010 in which fuzzy differential equations have been solved for tumor growth solutions [46,47].
Mahdieh in 2018 described a full fuzzy mathematical model of tumor growth with the help of IPD equations [48]. Souza et. al. [49] surfaced out the dynamics of tumor growth using fuzzy theory and fuzzy neotic threshold. Further investigation into fuzzy tumor modeling can be seen in [50 -54].

Algorithm and Architecture of the work
This work is aimed to check the behavior of tumor growth with the involvement of the Allee effect in a fuzzy environment with differentiability concepts (Generalized Hukuhara Derivative) in the population growth models. In this work, a fuzzy mathematical model is developed in which fuzziness is implemented in different parameters separately and together to check the tumor growth behaviour. Three cases are discussed as initial condition as fuzzy, coefficient as fuzzy and both as fuzzy in logistic equation with Allee effect in the tumour microenvironment. Numerical simulation is given to support this work. The algorithm and architecture of the proposed work is showen in Fig.1.

Triangular fuzzy number
A fuzzy number that is represented with three points as follows = ( , , ), ( < < ) and whose membership function is given by is called a triangular fuzzy number [55].

-cut
-cut of triangular fuzzy number = ( , , ) is given by

Generalized Hukuhara derivative for fuzzy valued function
The Generalized Hukuhara derivative for fuzzy valued function : ( , ) → ℜ ℱ at the point is defined as If ́( ∈ ℜ ℱ which can be found from (4), we can say that ( ) is generalized Hukuhara derivative at The main theory is that if ( ) is (i)-gH differentiable at then And ( ) is (ii)-gH differentiable at then Where the -cut of ( ) is defined as [ ( , ), ( , )] Proof: see the paper [56].

Characterization Theorem for Differential Equation in Fuzzy Environment
Consider the fuzzy initial value problem [57] as With initial condition ( ) = Proof: See paper [58] The parametric form of derivative of a fuzzy valued function is two types when the function is (i)-gH differentiable and (ii)-gH differentiable [59].

Crisp model Theory
Population evolution (cancer cells) are best described by Logistic population evolutionary models with the Allee effect as: With initial condition ( ) = Where is the number of cells, is the total growth rate, is carrying capacity, is a time of occurrence and is an Allee threshold. It is assumed that at time = 0, = 10 i.e. one billion cells. It is also assumed that < , 1 − is per capita logistic growth rate and 1 − ( − ) is modified per capita growth rate. However, in the presence of the Allee effect, the per capita growth decreases below a given population size and can be negative below the Allee threshold ( ) which eventually may extinct.

Proposed model of Tumor Growth in Fuzzy Environment
If we consider now equation (5) in the fuzzy environment then three cases arise in (5) as:

a) Fuzziness in the initial condition
The transformed model using differentiability concept, when Differentiable in equation (5

Results
This current work displayed the tumour growth model with Allee effect under fuzzy environment using Generalized Hukuhara Derivative method. The result represents the two cases simultaneously through both i-gH ( fig.2) and ii-gH ( fig.3) derivative for different values of ( = 0, 0.3, 0.7 1). Case I: It is analyzed that for strong Allee effect = 2, the cell number will grow and tends to the carrying capacity and eventually attaining stability if the initial population is greater than the Allee threshold. Case II: while as, the cell number will decline towards 0 and may exhibit extinction. if the initial population is smaller than the Allee threshold.
Hence, this all increase or decrease in cell number to a certain threshold makes it more appropriate to target the tumour treatment at early stages with suitable therapy.
The initial population and growth rate are greatly affected by the Allee effect as in the case of the higher cell population, there will be potentially high cooperative interaction and vice versa changes the growth rate accordingly. Fuzzy membership function clearly describes the mechanism of cancer growth with the Allee effect to a certain possibility degree. As in fig.2,3, at = 0.4, there is an increase in a cell population with an initial population greater than Allee threshold, and at = 0.3, there is decreasing trend of cell population with an initial population smaller than Allee threshold.

Discussion and Conclusion
For a better understanding of cancer dynamics, tumour growth modeling with the Allee effect in a fuzzy environment strongly confirms the occurrence of this ecological influence at the initiation of cancer. We explore the growth dynamics with the solution of the logistic equation Data availability: Not applicable.
Declarations: All the author have declared the manuscript for publication.
Ethics approval and consent to participate: This article does not contain any studies with human participants or animals.

Consent for publication:
All authors agree mutually with the participation and publication of the manuscript and declare that this is an original research.

Conflict of interest
The authors declare no competing interest.